Assignment #2
Agricultural and Applied Economics 637
(Due March 3, 2015)
(110 Points)
1. (20 pts) For this assignment I would like you to estimate a variety of
non-linear (in parameters) regression models and to undertake some postestimation analyses. To undertake this estimation I would like you to
modify the basic Gauss-Newton estimation algorithm we used in class to
add three new features. First, I would like you to add the capability
whereby you can choose between estimating a set of parameters via the:
(i) Gauss-Newton (GN) or (ii) Newton-Raphson (NR) method. When
using the GN method for estimation, use the GN method for obtaining
the parameter covariance matrix. Similarly use the NR covariance
method when estimating parameters via the NR algorithm. The choice of
estimation algorithm should be made by setting a control parameter in the
estimation launching program, e.g., Type = 1. Second add another
feature where if you use the GN algorithm for estimation, after
convergence, one more iteration will be undertaken using the NR
algorithm.
Third, add a feature to the program where, at the optimal parameter
values using either method you check for the convexity of the SSE
function. If it passes the test, print out a statement to that effect, calculate
parameter related statistics and then print out the traditional regression
output. If it does not pass the test then have the software print a message
to that effect and do not print out results. You can use this combined
code for the remainder of the assignment.
Apply this new code to the same type of Cobb-Douglas production
function you used in Assignment #1:
β
[1.1]
Quant t  γCapitalt Labortα + ε t
For this exercise I would like you to use Mizon’s (1977)’s data set. He
was one of the first authors to provide an overview on statistical
inference when using nonlinear regression models. In the 1977 paper he
estimated a variety of specifications for production functions. He used
U.K. data on capital, labor use and a common output measure for 24
industries for the years 1954, 1957 and 1960. The following table
provides a summary of the variables contained in the Mizon_1977.xls
dataset.
Table 1: Variables Contained in the Mizon (1977) Dataset
Variable
Description
Units
Quant Gross value-added at factor cost
Mil. $
Value of the stock of plant and
Capital
Mil. $.
machinery
Labor force available for work in the
LF
1,000
industry
Number of workers unemployed in the
Unemploy
1,000
industry
Average hours per week worked by those
Hour
Hours
employed
Year Survey data year
Year
#
Industry Industry ID Number
Assume that you would like to estimate the parameters of the above
Cobb-Douglas type production function where Labor is defined as the
number of worker-hours of labor input (divided by 100 for scaling
purposes). Present the typical regression output. [Note: You need to
calculate the labor variable using the above information.]
2. (45 pts) A problem faced by applied economists wanting to examine the
relationship between a dependent variable and a set of explanatory
variables is determination of the appropriate functional form. The BoxCox transformation is one method for letting the data determine the
functional form (Greene, p. 296-297). With the general relationship,
2
y=f(x1, x2,…xK), the Box-Cox transformation of the exogenous variables
x  1


x 
(x) can be represented via the following:

where x is strictly
positive, λ ≠ 0, k=(1,…,K), and λ is the parameter to estimate. As noted
in Greene (2008) p. 297 when λ = 0, via L’Hopital’s Rule:
x  1
 ln x . The functional form determined by the data is that

form defined by the estimated transformation parameter.
lim
 0
Consider the following equation, where the quantity of wool demanded,
Q, depends on the price of wool (PW) and on the price of synthetics (PS).
We can use the above Box-Cox transformation in the following demand
relationship:
ln  Q t   1 

  β P
β 2 PWtλ  1
3
λ
St
e
1
[2.1]
t
λ
λ
where β1, β2, β3, λ are unknown parameters and et is an iid random error
where et ~ (0, σ2).
(a) (10 pts) Use the WOOL data set and use NLS methods to estimate
the parameters for 3.1. Report the standard regression and
parameter-related statistics.
(b) (5 pts) Test the hypothesis that wool demand has a semi-log
functional form with respect to PW and PS.
(c) (5 pts) Test the hypothesis that β2 = − β3.
(d) (15 pts) Estimate the own and cross price elasticities of demand (i.e.,
∂lnQ/∂lnPW, ∂lnQ/∂lnPS) at the sample’s mean prices. Test whether
the own-price elasticities are different from −1. Is the cross-price
elasticity statistically different from 0? Test whether the own-price
elasticity is equal but of opposite sign of the cross-price elasticity.
(e) (10 pts) The above structure imposed the constraint that the
transformation parameter, λ, is the same for both PS and PW. Reestimate [3.1] via the following where this constraint is lifted:
ln  Qt   1 

λ
  β P
β2 PWt W  1
λW
3
λS
St
λS
e
1
t
[2.2]
3
where β1, β2, β3, λW, and λS are unknown parameters to be estimated.
Does the data provide statistical evidence that the transformation
parameters are indeed different from one another?
3. (45 pts) Researchers in the Department of Agronomy at the UW have
undertaken many years of fertilizer productivity studies using
experimental plot data. On the AAE637 web site is the spreadsheet file,
Corn_Plot_13.xls . This file contains plot data from crop year 2013.
This data contains information on corn output per acre (CYLD,
measured in bushels), and nitrogen application rates (N, measured in
pounds) and phosphorous application rates, P2O5 (P, measured in
pounds). Assume that you would like to estimate the parameters of the
following production function:
CYLD=f(N, P)
[3.1]
We obviously need to assume a functional form in order to evaluate the
structure of corn production. Assume you would like to use a version of
the Constant Elasticity of Substitution (CES) production function to
capture the above relationship. You decide to review the literature with
respect to the CES production and find an overview in the file
ces_2.pdf . You decide to estimate the parameters of the following CES
production function:

CYLD t  N 
 (1  )Pt   exp  e t  
t
[3.2]



-ρ 
2
ln  CYLD t    ln δN-ρ
t +(1-δ)Pt   e t e ~ 0,  IT


where: η ≡ the degree of homogeneity (scale parameter), η > 0
δ ≡ distribution parameter, 0 < δ < 1
ρ ≡ the substitution parameter, -1 < ρ < ∞, ρ ≠ 0


(a) (10 pts) Using the above 2013 plot data, estimate the 3 unknown
parameters, (ρ, δ, η) and associated standard errors.
(b) (5 pts) What is the correlation between predicted and actual values of
ln(CYLD)? In addition, what is the correlation between predicted
and actual values of CYLD using the parameters in [3.2]? [Note:
4
When evaluating the relationship between predicted and actual
values of CYLD (versus ln(CYLD)), you should note that the
E[exp(et)] need not equal 1.0 but in fact E[exp(et)] = exp(σ2/2). Use
the formula for estimating the value of σ2 as well as its parameter
induced variability].
(c) (5 pts) Does the production technology exhibit constant returns to
scale?
(d) (15 pts) Evaluate the marginal products of N and P when these
inputs are at their sample mean values. If using the CES_2.pdf for
your review of the CES production you should remember that in this
application α = 0 (i.e., the efficiency parameter in their
formulation). Are these marginal products positive from a statistical
point of view? [Note: Because of the nonlinear functional form, to
make sure you are actually evaluating the MP’s at a point on the
predicted nonlinear production function, use the predicted value of
CYLD based on average N and P values when evaluating the
marginal products. This implies that the variability of the
parameters associated in the calculation of the predicted quantity
should be explicitly included in the variance calculations of the
individual MP’s. As in (b) above, when evaluating the predicted
value of CYLD, the E[exp(et)] need not equal 1.0 but in fact
E[exp(et)] = exp(σ2/2). Use your estimate of σ2 and account for its
variance within the calculation of the MP’s.]
(e) (5 pts) At the mean values of the exogenous variables does the
Marginal Rate of Technical Substitution (MRTS) = 1.0?
(10 pts) Using the estimated production function parameters, you
would like to determine N and P application rates that minimize the
total cost of producing a desired corn yield, N* and P*. These
application rates can be obtained from the cost-minimizing
Lagrangian associated with [3.2]:
N*=N(PN,PP,CYLD)
[3.3]
P*=P(PN,PP,CYLD)
The general CES conditional (on CYLD) input demand functions
for N and P can be represented via the following:
5
1
 δ  1 ρ  
ρ
N*  CYLD 
 δPN

 PN 

 
1
1

1 ρ   1  δ  Pρ
P

1
1 ρ  


1
1
 1  δ  1 ρ  
ρ
1 ρ   1  δ  Pρ 1 ρ  
P*  CYLD 
δP
 N


P
 PP 


Assume a nitrogen price of $0.54/lb. and the price of P2O5 is
$0.20/lb. and CYLD is at the sample mean. What are the cost
minimizing application rates? What are their estimated standard
errors?
(f) (5 pts) Assume that corn yield is known and the data set’s mean
value. Test the null hypothesis that at this corn yield, the costminimizing fertilizer application rates, N* and P*, are equal (i.e., N*
= P*).
 


6